
where \(\mathrm {d}\mu =\star \,1\)  is the Riemannian measure defined by the
corresponding Hodge duality operator \(\star \,\) . The metric induces the Hodge operator
\(\star \,\colon \Omega ^2(\Sigma ;\operatorname{Ad}(P))\longrightarrow \Omega ^0(\Sigma ;\operatorname{Ad}(P))\)  acting only on the differential form part. We also equip the Lie
algebra \(\mathfrak {g}\)  with an invariant quadratic form, which we may
assume without loss of generality to be a suitable multiple, as described
in [1]}, of the Killing form on \(\mathfrak {g}\) 
denoted by \(\text{Tr}_{\mathfrak {g}}\) . Then Yang-Mills theory on \(\Sigma \)  is
defined by the \(\mathsf {Aut}(P)\) -invariant action functional of
\((P,A)\in \mathsf {Bun}_G^\nabla (\Sigma )\)  given by
\(S_{\textrm {\tiny YM}}(P,A)\frac{1}{4e^2}\,\int _\Sigma \,\text{Tr}_{\mathfrak {g}}\big (F_A\wedge \star \,F_A \big ) \ ,\) 
